 9819 R. Paul
 A KAM Theorem for Some Degenerate Hamiltonian Systems
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Jan 15, 98

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Abstract. We prove the existence of a large measure of invariant tori for
a class of perturbed integrable systems in which the unperturbed
Hamiltonian is degenerate (that is, its Hessian matrix does not
have full rank). This class consists of
systems for whichalong with certain other restrictionsthe unperturbed
Hamiltonian added to the average of the perturbation is nondegenerate.
Our result is similar to a theorem of Arnold, whose proof
is only sketched. In addition, we obtain explicit estimates on the measure
of phase space not foliated by invariant tori.
We apply our result to a system designed by Weinberg to test the
current theory of quantum mechanics. The system
is a perturbed simple harmonic oscillator, to which KAM
theorems with nondegeneracy conditions only on the unperturbed
Hamiltonian do not apply.
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